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[[Berkas:Matrix.svg|jmpl|247px|ka|Baris ''m'' adalah horizontal dan kolom ''n'' vertikal. Setiap elemen matriks sering dilambangkan menggunakan variabel dengan dua [[notasi indeks]]. Misalnya, ''a''<sub>2,1</sub> mewakili elemen pada baris kedua dan kolom pertama dari matriks '''A'''.]]
Dalam [[matematika]], '''matriks''' adalah [[wikt:susunan|susunan]]<ref>Secara ekuivalen, ''[[wikt:tabel|tabel]]''.</ref> [[bilangan]], [[simbol (formal)|simbol]], atau [[Ekspresi (matematika)|ekspresi]] yang disusun dalam [[wikt:baris|baris]] dan [[wikt:kolom|kolom]] sehingga membentuk suatu bangun [[persegi]].<ref>{{harvtxt|Anton|1987|p=23}}</ref><ref>{{harvtxt|Beauregard|Fraleigh|1973|p=56}}</ref> Sebagai contoh, dimensi matriks di bawah ini adalah matriks berukuran 2 × 3 (baca "dua kali tiga"):<math display="block">\begin{bmatrix}1 & 9 & -13 \\20 & 5 & -6 \end{bmatrix}</math>karena terdiri dari dua baris dan tiga kolom:.
:<math>\begin{bmatrix}1 & 9 & -13 \\20 & 5 & -6 \end{bmatrix}.</math>
Setiap objek dalam matriks '''<math>\mathbf{A}</math>''' berdimensi <math>m \times n</math> sering dilambangkan dengan <math>a_{i,j}</math>, dimana nilai maksimum <math>i = m</math> dan nilai maksimum <math>j = n</math>. Objek dalam matriks disebut ''elemen'', ''entri'', atau ''anggota'' matriks.<ref>{{cite book|last1=Young|first1=Cynthia|title=Precalculus|publisher=Laurie Rosatone|page=727|accessdate=2015-02-06}}</ref>
== Notasi ==
Matriks pada umumnya ditulis dalam tanda kurung siku/kurung kurawal:<math display="block"> \mathbf{A} =
<math> \mathbf{A} =
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{pmatrix}=\left(a_{ij}\right) \in \mathbb{R}^{m \times n}.
</math>Notasi simbolik untuk menyatakan suatu matriks sangat bervariasi, namun beberapa notasi lebih umum dipakai. Matriks biasanya dilambangkan dengan menggunakan huruf besar (seperti '''<math>\mathbf{A}</math>''' pada contoh di atas). Sedangkan huruf kecil yang sesuai, dengan dua indeks subskrip, misal <math>a_{1,1}</math>, untuk menyebutkan elemen matriks tersebut. Selain menggunakan huruf besar untuk melambangkan matriks, banyak penulis menggunakan gaya tipografi khusus, yang biasanya dicetak tebal tegak, untuk lebih membedakan matriks dari objek matematika lainnya. Notasi alternatif melibatkan penggunaan ''double-underline'' (garis bawah ganda) dengan nama variabel, dengan atau tanpa gaya cetak tebal (contohnya <math>\underline{\underline{A}}</math>).
</math>
Spesifikasi notasi simbolik dari matriks sangat bervariasi, dengan beberapa tren yang umum dipakai. Matriks biasanya dilambangkan dengan menggunakan huruf besar (seperti '''<math>\mathbf{A}</math>''' pada contoh di atas). Sedangkan huruf kecil yang sesuai, dengan dua indeks subskrip, misal <math>a_{1,1}</math>, untuk menyebutkan elemen matriks tersebut. Selain menggunakan huruf besar untuk melambangkan matriks, banyak penulis menggunakan gaya tipografi khusus, yang biasanya dicetak tebal tegak, untuk lebih membedakan matriks dari objek matematika lainnya. Notasi alternatif melibatkan penggunaan ''double-underline'' (garis bawah ganda) dengan nama variabel, dengan atau tanpa gaya cetak tebal (contohnya <math>\underline{\underline{A}}</math>).
Elemen baris ke-<math>i</math> dan kolom ke-<math>j</math> dari matriks '''<math>\mathbf{A}</math>''' terkadang dirujuk sebagai elemen ke <math>(i,\,j)</math>dari matriks, dan umumnya ditulis sebagai <math>a_{i,\,j}</math>atau <math>a_{ij}</math>. Alternatif notasi yang lain adalah <math>A[i,j]</math> atau <math>A_{i,j}</math>. Sebagai contoh, elemen ke <math>(1, 3)</math> dari matriks '''<math>\mathbf{A}</math>''' berikut dapat ditulis sebagai <math>a_{1,\,3
# [[Matriks diagonal]]: merupakan matriks persegi yang <math>a_{ij}=0</math>, untuk <math>i \neq j</math>
# Matriks skalar: merupakan matriks diagonal yang memiliki unsur [[Diagonal utama|diagonal utamanya]] sama, misalnya <math>k</math>
# [[Matriks identitas]]: merupakan matriks skalar di mana <math>k=1</math>
# [[Matriks simetrissimetrik]]: merupakan matriks persegi dengan <math>a_{ij}=a_{ji}</math> untuk <math>\forall_{i,j}</math>.
# Matriks anti simetris: merupakan matriks persegi yang transposenya adalah negatif dari matriks tersebut dengan <math>a_{ij}= -a_{ji}</math>
# [[Matriks segitiga|Matriks segitiga atas]]: merupakan matriks persegi yang semua unsurnya dibawah diagonal utamanya adalah 0, yakni <math>a_{ij}= 0</math> ketika <math>i>j</math>
b_{n1}&&\dotsc&&\dotsc&&b_{nn} \end{bmatrix}</math>
Hasil pertambahan dua matriks tersebut yaitu <math display="block">A+B = \begin{bmatrix}a_{11} + b_{11}&&a_{12}+b_{12}&&\dotsc&&a_{1n} + b_{1n}\\
a_{21}+b_{21}&&\ddots&&\cdots&&\vdots\\
\vdots&&\cdots&&\ddots&&\vdots\\
Perhatikan bahwa, elemen-elemen pada hasil operasi pertamahan matriks tersebut tidak lain merupakan penjumlahan pada suatu bilangan dan berlaku sifat komutatif, <math>a_{11}+b_{11} = b_{11} + a_{11}</math>, dengan demikian dapat dituliskan sebagai
<math>\left[\begin{matrix}b_{11} + a_{11}&&b_{12}+a_{12}&&\dotsc&&b_{1n} + a_{1n}\\
==== Matriks diagonal dan segitiga ====
Jika semua entri <span id="linear_maps">'''<math>\mathbf{A}</math>'''</span> di bawah diagonal utama adalah nol, <span id="linear_maps">'''<math>\mathbf{A}</math>'''</span> disebut ''[[matriks segitiga]] atas''. Demikian pula jika semua entri ''<span id="linear_maps">'''<math>\mathbf{A}</math>'''</span>'' di atas diagonal utama adalah nol, <span id="linear_maps">'''<math>\mathbf{A}</math>'''</span> disebut ''matriks segitiga bawah''. Jika semua entri di luar diagonal utama adalah nol, <span id="linear_maps">'''<math>\mathbf{A}</math>'''</span> disebut [[matriks diagonal]].
<!--
==== Matriks identitas ====
{{Main|Matriks identitas}}
The ''identity matrix'' '''I'''{{sub|''n''}} of size ''n'' is the ''n''-by-''n'' matrix in which all the elements on the [[main diagonal]] are equal to 1 and all other elements are equal to 0, for example,
:<math>
\mathbf{I}_1 = \begin{bmatrix} 1 \end{bmatrix},
\ \mathbf{I}_2 = \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix},
\ \cdots ,
\ \mathbf{I}_n = \begin{bmatrix}
1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 1
\end{bmatrix}
</math>
It is a square matrix of order ''n'', and also a special kind of [[diagonal matrix]]. It is called an identity matrix because multiplication with it leaves a matrix unchanged:
:{{nowrap begin}}'''AI'''{{sub|''n''}} = '''I'''{{sub|''m''}}'''A''' = '''A'''{{nowrap end}} for any ''m''-by-''n'' matrix '''A'''.
A nonzero scalar multiple of an identity matrix is called a ''scalar'' matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group of nonzero elements of the field.
====Symmetric or skew-symmetric matrix====
A square matrix '''A''' that is equal to its transpose, that is, {{nowrap begin}}'''A''' = '''A'''{{sup|T}}{{nowrap end}}, is a [[symmetric matrix]]. If instead, '''A''' is equal to the negative of its transpose, that is, {{nowrap begin}}'''A''' = −'''A'''{{sup|T}},{{nowrap end}} then '''A''' is a [[skew-symmetric matrix]]. In complex matrices, symmetry is often replaced by the concept of [[Hermitian matrix|Hermitian matrices]], which satisfy '''A'''{{sup|∗}} = '''A''', where the star or [[asterisk]] denotes the [[conjugate transpose]] of the matrix, that is, the transpose of the [[complex conjugate]] of '''A'''.
By the [[spectral theorem]], real symmetric matrices and complex Hermitian matrices have an [[eigenbasis]]; that is, every vector is expressible as a [[linear combination]] of eigenvectors. In both cases, all eigenvalues are real.<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Theorem 2.5.6}}</ref> This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see [[#Infinite matrices|below]].
====Invertible matrix and its inverse====
A square matrix '''A''' is called ''[[invertible matrix|invertible]]'' or ''non-singular'' if there exists a matrix '''B''' such that
:'''AB''' = '''BA''' = '''I'''{{sub|''n''}} ,<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition I.2.28}}</ref><ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition I.5.13}}</ref>
where '''I'''{{sub|''n''}} is the ''n''×''n'' [[identity matrix]] with 1s on the [[main diagonal]] and 0s elsewhere. If '''B''' exists, it is unique and is called the ''[[Invertible matrix|inverse matrix]]'' of '''A''', denoted '''A'''{{sup|−1}}.
====Definite matrix====
{| class="wikitable" style="float:right; text-align:center; margin:0ex 0ex 2ex 2ex;"
|-
! [[Positive definite matrix]] !! [[Indefinite matrix]]
|-
| <math>\begin{bmatrix}
\frac{1}{4} & 0 \\
0 & 1 \\
\end{bmatrix}</math>
| <math>\begin{bmatrix}
\frac{1}{4} & 0 \\
0 & -\frac{1}{4}
\end{bmatrix}</math>
|-
| ''Q''(''x'', ''y'') = 1/4 ''x''{{sup|2}} + ''y''{{sup|2}}
| ''Q''(''x'', ''y'') = 1/4 ''x''{{sup|2}} − 1/4 ''y''{{sup|2}}
|-
| [[File:Ellipse in coordinate system with semi-axes labelled.svg|150px]] <br>Points such that ''Q''(''x'',''y'')=1 <br> ([[Ellipse]]).
| [[File:Hyperbola2 SVG.svg|150px]] <br> Points such that ''Q''(''x'',''y'')=1 <br> ([[Hyperbola]]).
|}
A symmetric ''n''×''n''-matrix '''A''' is called [[positive-definite matrix|''positive-definite'']] if the associated [[quadratic form]]
:<span id="quadratic forms">''f''{{spaces|hair}}('''x''') = '''x'''{{sup|T}}'''A{{nbsp}}x'''</span>
has a positive value for every nonzero vector '''x''' in '''R'''{{sup|''n''}}. If ''f''{{spaces|hair}}('''x''') only yields negative values then '''A''' is [[definiteness of a matrix#Negative definite|''negative-definite'']]; if ''f'' does produce both negative and positive values then '''A''' is [[definiteness of a matrix#Indefinite|''indefinite'']].<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Chapter 7}}</ref> If the quadratic form ''f'' yields only non-negative values (positive or zero), the symmetric matrix is called ''positive-semidefinite'' (or if only non-positive values, then negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.
A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is, the matrix is positive-semidefinite and it is invertible.<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Theorem 7.2.1}}</ref> The table at the right shows two possibilities for 2-by-2 matrices.
Allowing as input two different vectors instead yields the [[bilinear form]] associated to '''A''':
:''B''{{sub|'''A'''}} ('''x''', '''y''') = '''x'''{{sup|T}}'''Ay'''.<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Example 4.0.6, p. 169}}</ref>
====Orthogonal matrix====
{{Main|Orthogonal matrix}}
An ''orthogonal matrix'' is a [[#Square matrices|square matrix]] with [[real number|real]] entries whose columns and rows are [[orthogonal]] [[unit vector]]s (that is, [[orthonormality|orthonormal]] vectors). Equivalently, a matrix '''A''' is orthogonal if its [[transpose]] is equal to its [[invertible matrix|inverse]]:
:<math>\mathbf{A}^\mathrm{T}=\mathbf{A}^{-1}, \,</math>
which entails
:<math>\mathbf{A}^\mathrm{T} \mathbf{A} = \mathbf{A} \mathbf{A}^\mathrm{T} = \mathbf{I}_n,</math>
where '''I'''{{sub|''n''}} is the [[identity matrix]] of size ''n''.
An orthogonal matrix '''A''' is necessarily [[invertible matrix|invertible]] (with inverse {{nowrap|1='''A'''{{sup|−1}} = '''A'''{{sup|T}}}}), [[unitary matrix|unitary]] ({{nowrap|1='''A'''{{sup|−1}} = '''A'''*}}), and [[normal matrix|normal]] ({{nowrap|1='''A'''*'''A''' = '''AA'''*}}). The [[determinant]] of any orthogonal matrix is either {{math|+1}} or {{math|−1}}. A ''special orthogonal matrix'' is an orthogonal matrix with [[determinant]] +1. As a [[linear transformation]], every orthogonal matrix with determinant {{math|+1}} is a pure [[rotation (mathematics)|rotation]] without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant {{math|-1}} reverses the orientation, i.e., is a composition of a pure [[reflection (mathematics)|reflection]] and a (possibly null) rotation. The identity matrices have determinant {{math|1}}, and are pure rotations by an angle zero.
The [[complex number|complex]] analogue of an orthogonal matrix is a [[unitary matrix]].
===Main operations===
====Trace====
The [[trace of a matrix|trace]], tr('''A''') of a square matrix '''A''' is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned [[#non commutative|above]], the trace of the product of two matrices is independent of the order of the factors:
: tr('''AB''') = tr('''BA''').
This is immediate from the definition of matrix multiplication:
:<math>\operatorname{tr}(\mathbf{AB}) = \sum_{i=1}^m \sum_{j=1}^n a_{ij} b_{ji} = \operatorname{tr}(\mathbf{BA}).</math>
It follows that the trace of the product of more than two matrices is independent of [[cyclic permutation]]s of the matrices, however this does not in general apply for arbitrary permutations (for example, tr('''ABC''') ≠ tr('''BAC'''), in general). Also, the trace of a matrix is equal to that of its transpose, that is,
:{{nowrap begin}}tr('''A''') = tr('''A'''{{sup|T}}){{nowrap end}}.
====Determinant====
{{Main|Determinant}}
[[File:Determinant example.svg|thumb|300px|right|A linear transformation on '''R'''{{sup|2}} given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the [[orientation (mathematics)|orientation]], since it turns the counterclockwise orientation of the vectors to a clockwise one.]]
The ''determinant'' of a square matrix '''A''' (denoted det('''A''') or |'''A'''|<ref name=":2" />) is a number encoding certain properties of the matrix. A matrix is invertible [[if and only if]] its determinant is nonzero. Its [[absolute value]] equals the area (in '''R'''{{sup|2}}) or volume (in '''R'''{{sup|3}}) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.
The determinant of 2-by-2 matrices is given by
:<math>\det \begin{bmatrix}a&b\\c&d\end{bmatrix} = ad-bc.</math><ref name=":3" />
The determinant of 3-by-3 matrices involves 6 terms ([[rule of Sarrus]]). The more lengthy [[Leibniz formula for determinants|Leibniz formula]] generalises these two formulae to all dimensions.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition III.2.1}}</ref>
The determinant of a product of square matrices equals the product of their determinants:
:{{nowrap begin}}det('''AB''') = det('''A''') · det('''B''').<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Theorem III.2.12}}</ref>{{nowrap end}}
Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Corollary III.2.16}}</ref> Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the [[Laplace expansion]] expresses the determinant in terms of [[minor (linear algebra)|minors]], that is, determinants of smaller matrices.<ref>{{Harvard citations |last1=Mirsky |year=1990 |nb=yes |loc=Theorem 1.4.1}}</ref> This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve [[linear system]]s using [[Cramer's rule]], where the division of the determinants of two related square matrices equates to the value of each of the system's variables.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Theorem III.3.18}}</ref>
====Eigenvalues and eigenvectors====
{{Main|Eigenvalue, eigenvector and eigenspace|l1=Eigenvalues and eigenvectors}}
A number λ and a non-zero vector '''v''' satisfying
:'''Av''' = λ'''v'''
are called an ''eigenvalue'' and an ''eigenvector'' of '''A''', respectively.<ref>''Eigen'' means "own" in [[German language|German]] and in [[Dutch language|Dutch]].</ref><ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition III.4.1}}</ref> The number λ is an eigenvalue of an ''n''×''n''-matrix '''A''' if and only if '''A'''−λ'''I'''{{sub|''n''}} is not invertible, which is [[logical equivalence|equivalent]] to
:<math>\det(\mathsf{A}-\lambda \mathsf{I}) = 0.</math><ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition III.4.9}}</ref>
The polynomial ''p''{{sub|'''A'''}} in an [[indeterminate (variable)|indeterminate]] ''X'' given by evaluation of the determinant det(''X'''''I'''{{sub|''n''}}−'''A''') is called the [[characteristic polynomial]] of '''A'''. It is a [[monic polynomial]] of [[degree of a polynomial|degree]] ''n''. Therefore the polynomial equation ''p''{{sub|'''A'''}}(λ){{nbsp}}={{nbsp}}0 has at most ''n'' different solutions, that is, eigenvalues of the matrix.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Corollary III.4.10}}</ref> They may be complex even if the entries of '''A''' are real. According to the [[Cayley–Hamilton theorem]], {{nowrap begin}}''p''{{sub|'''A'''}}('''A''') = '''0'''{{nowrap end}}, that is, the result of substituting the matrix itself into its own characteristic polynomial yields the [[zero matrix]].-->
== Lihat pula ==
* [[Aljabar linear]]
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